DESCRIPTION (Applicant's abstract): In statistical modeling the covariance structure is often considered a nuisance, or at least of secondary importance to the mean. However, when the goals of an analysis include estimation of subject-specific effects or prediction, estimation of the covariance structure is very important. This project will develop methodology for improving estimation of covariance structure in longitudinal cancer studies. This will result in increased efficiency in estimation of fixed and random effects, such as subject-specific trajectories, and predictions. Specifically, this project will develop estimators of covariance matrices that are robust to a variety of eigenstructures and/or structural assumptions, develop methods to compute these estimators in hierarchical models, and develop classes of models to account for heterogeneity, both explained and unexplained by covariates, in covariance structures across subjects in longitudinal trials. In sum, this will provide greater flexibility in modeling, and efficiency in making inferences from, longitudinal cancer data. The methods include development of sensible prior distribution from which estimators can be derived that have good properties in small samples and/or in high dimensions and construction of hierarchical models to account for heterogeneous covariance structures across subjects through covariates and/or prior distributions. The common theme will be prior distributions which sh6nk the cova6ance matrix or function toward some parametric form or to some 'average' matrix or function, with the amount of shrinkage determined by the data. Computationally efficient ways to compute the estimators and fit the models will be explored, some of which will involve only minor modifications to standard software.